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Today in my Analysis class my teacher proved that (real)symmetric bilinear forms (in an inner-product space) are orthogonally diagonalizable using compactness, differentiablity of innerproducts and other concepts of analysis and then he proceeded to prove that the k'th largest eigenvalue can be derived by $$\min_{W\subset V,\dim W=k} \max_{x\in W , \|x\|=1} \langle Ax,x\rangle$$ where the bilinear form is $B(x,y)=\langle Ax,y\rangle$.

Can anyone suggest me some books which have these kind of "analysis" flavoured linear algebra?

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You can try Hubbard and Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms."

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